Operad

Operads originate in algebraic topology; they were introduced to characterize iterated loop spaces by J. Michael Boardman and Rainer M. Vogt in 1968{{Cite journal|last1=Boardman|first1=J. M.|author-link=Michael Boardman|last2=Vogt|first2=R. M.|date=1 November 1968|title=Homotopy-everything $H$-spaces|url=https://www.ams.org/journals/bull/1968-74-06/S0002-9904-1968-12070-1/home.html|journal=Bulletin of the American Mathematical Society|language=en-US|volume=74|issue=6|pages=1117–1123|doi=10.1090/S0002-9904-1968-12070-1|issn=0002-9904|doi-access=free}}{{Cite book|last1=Boardman|first1=J. M.|author-link=Michael Boardman|last2=Vogt|first2=R. M.|date=1973|title=Homotopy Invariant Algebraic Structures on Topological Spaces|series=Lecture Notes in Mathematics|language=en-gb|volume=347|doi=10.1007/bfb0068547|issn=0075-8434|isbn=978-3-540-06479-4}} and by J. Peter May in 1972.{{Cite book|last=May|first=J. P.|author-link=J. Peter May|date=1972|title=The Geometry of Iterated Loop Spaces|series=Lecture Notes in Mathematics|language=en-gb|volume=271|doi=10.1007/bfb0067491|issn=0075-8434|isbn=978-3-540-05904-2|citeseerx=10.1.1.146.3172}}

Martin Markl, Steve Shnider, and Jim Stasheff write in their book on operads:"Operads in Algebra, Topology and Physics": Martin Markl, Steve Shnider, Jim Stasheff, Mathematical Surveys and Monographs, Volume: 96; 2002

: "The name operad and the formal definition appear first in the early 1970's in J. Peter May's "The Geometry of Iterated Loop Spaces", but a year or more earlier, Boardman and Vogt described the same concept under the name categories of operators in standard form, inspired by PROPs and PACTs of Adams and Mac Lane. In fact, there is an abundance of prehistory. Weibel [Wei] points out that the concept first arose a century ago in A.N. Whitehead's "A Treatise on Universal Algebra", published in 1898."

The word "operad" was created by May as a portmanteau of "operations" and "monad" (and also because his mother was an opera singer).{{Cite web|url=https://www.math.uchicago.edu/~may/PAPERS/mayi.pdf|title=Operads, Algebras, and Modules|last=May|first=J. Peter|author-link=J. Peter May|website=math.uchicago.edu|page=2|access-date=28 September 2018}}

Interest in operads was considerably renewed in the early 90s when, based on early insights of Maxim Kontsevich, Victor Ginzburg and Mikhail Kapranov discovered that some duality phenomena in rational homotopy theory could be explained using Koszul duality of operads.{{Cite journal|last1=Ginzburg|first1=Victor|author-link=Victor Ginzburg|last2=Kapranov|first2=Mikhail|date=1994|title=Koszul duality for operads|url=https://projecteuclid.org/euclid.dmj/1077286744|journal=Duke Mathematical Journal|language=en|volume=76|issue=1|pages=203–272|doi=10.1215/S0012-7094-94-07608-4|issn=0012-7094|mr=1301191|zbl=0855.18006|s2cid=115166937|via=Project Euclid}}{{Cite web|url=http://www.numdam.org/item/SB_1994-1995__37__47_0|title=La renaissance des opérades|last=Loday|first=Jean-Louis|author-link=Jean-Louis Loday|year=1996|website=www.numdam.org|series=Séminaire Nicolas Bourbaki|language=en|mr=1423619|zbl=0866.18007|access-date=27 September 2018}} Operads have since found many applications, such as in deformation quantization of Poisson manifolds, the Deligne conjecture,{{cite arXiv|last1=Kontsevich|first1=Maxim|last2=Soibelman|first2=Yan|date=26 January 2000|title=Deformations of algebras over operads and Deligne's conjecture|eprint=math/0001151}} or graph homology in the work of Maxim Kontsevich and Thomas Willwacher.

Suppose $X$ is a set and for $n\backslash in\backslash N$ we define

:$P(n):=\backslash \{f\backslash colon\; X^n\backslash to\; X\backslash \}$,

the set of all functions from the cartesian product of $n$ copies of $X$ to $X$.

We can compose these functions: given $f\backslash in\; P(n)$, $f\_1\backslash in\; P(k\_1),\backslash ldots,f\_n\backslash in\; P(k\_n)$, the function

:$f\; \backslash circ\; (f\_1,\backslash ldots,f\_n)\backslash in\; P(k\_1+\backslash cdots+k\_n)$

is defined as follows: given $k\_1+\backslash cdots+k\_n$ arguments from $X$, we divide them into $n$ blocks, the first one having $k\_1$ arguments, the second one $k\_2$ arguments, etc., and then apply $f\_1$ to the first block, $f\_2$ to the second block, etc. We then apply $f$ to the list of $n$ values obtained from $X$ in such a way.

We can also permute arguments, i.e. we have a right action $*$ of the symmetric group $S\_n$ on $P(n)$, defined by

:$(f*s)(x\_1,\backslash ldots,x\_n)\; =\; f(x\_\{s^\{-1\}(1)\},\backslash ldots,x\_\{s^\{-1\}(n)\})$

for $f\backslash in\; P(n)$, $s\backslash in\; S\_n$ and $x\_1,\backslash ldots,x\_n\backslash in\; X$.

The definition of a symmetric operad given below captures the essential properties of these two operations $\backslash circ$ and $*$.

A non-symmetric operad (sometimes called an operad without permutations, or a non-$\backslash Sigma$ or plain operad) consists of the following:

• a sequence $(P(n))\_\{n\backslash in\backslash mathbb\{N\}\}$ of sets, whose elements are called $n$-ary operations,
• an element $1$ in $P(1)$ called the identity,
• for all positive integers $n$, $k\_1,\backslash ldots,k\_n$, a composition function

:$$

\begin{align}

\circ: P(n)\times P(k_1)\times\cdots\times P(k_n) & \to P(k_1+\cdots+k_n)\\

(\theta,\theta_1,\ldots,\theta_n) & \mapsto \theta\circ(\theta_1,\ldots,\theta_n),

\end{align}

satisfying the following coherence axioms:

• identity: $\backslash theta\backslash circ(1,\backslash ldots,1)=\backslash theta=1\backslash circ\backslash theta$
• associativity:

:: $$

\begin{align}

& \theta \circ \Big(\theta_1 \circ (\theta_{1,1}, \ldots, \theta_{1,k_1}), \ldots, \theta_n \circ (\theta_{n,1}, \ldots,\theta_{n,k_n})\Big) \\

= {} & \Big(\theta \circ (\theta_1, \ldots, \theta_n)\Big) \circ (\theta_{1,1}, \ldots, \theta_{1,k_1}, \ldots, \theta_{n,1}, \ldots, \theta_{n,k_n})

\end{align}

A symmetric operad (often just called operad) is a non-symmetric operad $P$ as above, together with a right action of the symmetric group $S\_n$ on $P(n)$ for $n\backslash in\backslash N$, denoted by $*$ and satisfying

• equivariance: given a permutation $t\backslash in\; S\_n$,

:: $$

(\theta*t)\circ(\theta_{1},\ldots,\theta_{n}) = (\theta\circ(\theta_{t^{-1}(1)},\ldots,\theta_{t^{-1}(n)}))*t'

::(where $t\text{'}$ on the right hand side refers to the element of $S\_\{k\_1+\backslash dots+k\_n\}$ that acts on the set $\backslash \{1,\; 2,\; \backslash dots\; ,\; k\_1+\backslash dots\; +k\_n\backslash \}$ by breaking it into $n$ blocks, the first of size $k\_1$, the second of size $k\_2$, through the $n$th block of size $k\_n$, and then permutes these $n$ blocks by $t$, keeping each block intact)

:and given $n$ permutations $s\_i\; \backslash in\; S\_\{k\_i\}$,

::$$

\theta\circ(\theta_1*s_1,\ldots,\theta_n*s_n) = (\theta\circ(\theta_1,\ldots,\theta_n))*(s_1,\ldots,s_n)

::(where $(s\_1,\backslash ldots,s\_n)$ denotes the element of $S\_\{k\_1+\backslash dots+k\_n\}$ that permutes the first of these blocks by $s\_1$, the second by $s\_2$, etc., and keeps their overall order intact).

The permutation actions in this definition are vital to most applications, including the original application to loop spaces.

A morphism of operads $f:P\backslash to\; Q$ consists of a sequence

:$(f\_n:P(n)\backslash to\; Q(n))\_\{n\backslash in\backslash mathbb\{N\}\}$

that:

• preserves the identity: $f(1)=1$
• preserves composition: for every n-ary operation $\backslash theta$ and operations $\backslash theta\_1\; ,\; \backslash ldots\; ,\; \backslash theta\_n$,

:: $$

f(\theta\circ(\theta_1,\ldots,\theta_n))

= f(\theta)\circ(f(\theta_1),\ldots,f(\theta_n))

• preserves the permutation actions: $f(x*s)=f(x)*s$.

Operads therefore form a category denoted by $\backslash mathsf\{Oper\}$.

So far operads have only been considered in the category of sets. More generally, it is possible to define operads in any symmetric monoidal category C . In that case, each $P(n)$ is an object of C, the composition $\backslash circ$ is a morphism $P(n)\backslash otimes\; P(k\_1)\backslash otimes\backslash cdots\backslash otimes\; P(k\_n)\; \backslash to\; P(k\_1+\backslash cdots+k\_n)$ in C (where $\backslash otimes$ denotes the tensor product of the monoidal category), and the actions of the symmetric group elements are given by isomorphisms in C.

A common example is the category of topological spaces and continuous maps, with the monoidal product given by the cartesian product. In this case, an operad is given by a sequence of spaces (instead of sets) $\backslash \{\; P(n)\; \backslash \}\_\{n\; \backslash ge\; 0\}$. The structure maps of the operad (the composition and the actions of the symmetric groups) are then assumed to be continuous. The result is called a topological operad. Similarly, in the definition of a morphism of operads, it would be necessary to assume that the maps involved are continuous.

Other common settings to define operads include, for example, modules over a commutative ring, chain complexes, groupoids (or even the category of categories itself), coalgebras, etc.

Given a commutative ring R we consider the category $R\backslash text\{-\}\backslash mathsf\{Mod\}$ of modules over R. An operad over R can be defined as a monoid object $(T,\; \backslash gamma,\; \backslash eta)$ in the monoidal category of endofunctors on $R\backslash text\{-\}\backslash mathsf\{Mod\}$ (it is a monad) satisfying some finiteness condition.”finiteness" refers to the fact that only a finite number of inputs are allowed in the definition of an operad. For example, the condition is satisfied if one can write

:$T(V)\; =\; \backslash bigoplus\_\{n=1\}^\{\backslash infty\}\; T\_n\; \backslash otimes\; V^\{\backslash otimes\; n\}$,

:$\backslash gamma(V):\; T\_n\; \backslash otimes\; T\_\{i\_1\}\; \backslash otimes\; \backslash cdots\; \backslash otimes\; T\_\{i\_n\}\; \backslash to\; T\_\{i\_1\; +\; \backslash dots\; +\; i\_n\}$.

For example, a monoid object in the category of "polynomial endofunctors" on $R\backslash text\{-\}\backslash mathsf\{Mod\}$ is an operad. Similarly, a symmetric operad can be defined as a monoid object in the category of $\backslash mathbb\{S\}$-objects, where $\backslash mathbb\{S\}$ means a symmetric group.{{cite arXiv|last1=Jones|first1=J. D. S.|last2=Getzler|first2=Ezra|date=8 March 1994|title=Operads, homotopy algebra and iterated integrals for double loop spaces|eprint=hep-th/9403055|language=en}} A monoid object in the category of combinatorial species is an operad in finite sets.

An operad in the above sense is sometimes thought of as a generalized ring. For example, Nikolai Durov defines his generalized rings as monoid objects in the monoidal category of endofunctors on $\backslash textbf\{Set\}$ that commute with filtered colimits.N. Durov, New approach to Arakelov geometry, University of Bonn, PhD thesis, 2007; [http://www.arxiv.org/abs/0704.2030 arXiv:0704.2030]. This is a generalization of a ring since each ordinary ring R defines a monad $\backslash Sigma\_R:\; \backslash textbf\{Set\}\; \backslash to\; \backslash textbf\{Set\}$ that sends a set X to the underlying set of the free R-module $R^\{(X)\}$ generated by X.

"Associativity" means that composition of operations is associative

(the function $\backslash circ$ is associative), analogous to the axiom in category theory that $f\; \backslash circ\; (g\; \backslash circ\; h)\; =\; (f\; \backslash circ\; g)\; \backslash circ\; h$; it does not mean that the operations themselves are associative as operations.

Compare with the associative operad, below.

Associativity in operad theory means that expressions can be written involving operations without ambiguity from the omitted compositions, just as associativity for operations allows products to be written without ambiguity from the omitted parentheses.

For instance, if $\backslash theta$ is a binary operation, which is written as $\backslash theta(a,b)$ or $(ab)$. So that $\backslash theta$ may or may not be associative.

Then what is commonly written $((ab)c)$ is unambiguously written operadically as $\backslash theta\; \backslash circ\; (\backslash theta,1)$ . This sends $(a,b,c)$ to $(ab,c)$ (apply $\backslash theta$ on the first two, and the identity on the third), and then the $\backslash theta$ on the left "multiplies" $ab$ by $c$.

This is clearer when depicted as a tree:

Tree before composition

which yields a 3-ary operation:

Tree after composition

{{Clear}}

However, the expression $(((ab)c)d)$ is a priori ambiguous:

it could mean $\backslash theta\; \backslash circ\; ((\backslash theta,1)\; \backslash circ\; ((\backslash theta,1),1))$, if the inner compositions are performed first, or it could mean $(\backslash theta\; \backslash circ\; (\backslash theta,1))\; \backslash circ\; ((\backslash theta,1),1)$,

if the outer compositions are performed first (operations are read from right to left).

Writing $x=\backslash theta,\; y=(\backslash theta,1),\; z=((\backslash theta,1),1)$, this is $x\; \backslash circ\; (y\; \backslash circ\; z)$ versus $(x\; \backslash circ\; y)\; \backslash circ\; z$. That is, the tree is missing "vertical parentheses":

Tree before composition

If the top two rows of operations are composed first (puts an upward parenthesis at the $(ab)c\backslash \; \backslash \; d$ line; does the inner composition first), the following results:

Intermediate tree

which then evaluates unambiguously to yield a 4-ary operation.

As an annotated expression:

:$\backslash theta\_\{(ab)c\backslash cdot\; d\}\; \backslash circ\; ((\backslash theta\_\{ab\; \backslash cdot\; c\},1\_d)\; \backslash circ\; ((\backslash theta\_\{a\backslash cdot\; b\},1\_c),1\_d))$

Tree after composition

If the bottom two rows of operations are composed first (puts a downward parenthesis at the $ab\backslash quad\; c\backslash \; \backslash \; d$ line; does the outer composition first), following results:

Intermediate tree

which then evaluates unambiguously to yield a 4-ary operation:

Tree after composition

The operad axiom of associativity is that these yield the same result, and thus that the expression $(((ab)c)d)$ is unambiguous.

The identity axiom (for a binary operation) can be visualized in a tree as:

The axiom of identity in an operad

meaning that the three operations obtained are equal: pre- or post- composing with the identity makes no difference. As for categories, $1\; \backslash circ\; 1\; =\; 1$ is a corollary of the identity axiom.

The most basic operads are the ones given in the section on "Intuition", above. For any set $X$, we obtain the endomorphism operad $\backslash mathcal\{End\}\_X$ consisting of all functions $X^n\backslash to\; X$. These operads are important because they serve to define operad algebras. If $\backslash mathcal\{O\}$ is an operad, an operad algebra over $\backslash mathcal\{O\}$ is given by a set $X$ and an operad morphism $\backslash mathcal\{O\}\; \backslash to\; \backslash mathcal\{End\}\_X$. Intuitively, such a morphism turns each "abstract" operation of $\backslash mathcal\{O\}(n)$ into a "concrete" $n$-ary operation on the set $X$. An operad algebra over $\backslash mathcal\{O\}$ thus consists of a set $X$ together with concrete operations on $X$ that follow the rules abstractely specified by the operad $\backslash mathcal\{O\}$.

If k is a field, we can consider the category of finite-dimensional vector spaces over k; this becomes a monoidal category using the ordinary tensor product over k. We can then define endomorphism operads in this category, as follows. Let V be a finite-dimensional vector space The endomorphism operad $\backslash mathcal\{End\}\_V\; =\; \backslash \{\; \backslash mathcal\{End\}\_V(n)\; \backslash \}$ of V consists of{{cite journal|last1=Markl|first1=Martin|year=2006|title=Operads and PROPs|journal=Handbook of Algebra|volume=5|issue=1|pages=87–140|doi=10.1016/S1570-7954(07)05002-4|arxiv=math/0601129 |isbn=9780444531018|s2cid=3239126}} Example 2

1. $\backslash mathcal\{End\}\_V(n)$ = the space of linear maps $V^\{\backslash otimes\; n\}\; \backslash to\; V$,
2. (composition) given $f\backslash in\backslash mathcal\{End\}\_V(n)$, $g\_1\backslash in\backslash mathcal\{End\}\_V(k\_1)$, ..., $g\_n\backslash in\backslash mathcal\{End\}\_V(k\_n)$, their composition is given by the map $V^\{\backslash otimes\; k\_1\}\; \backslash otimes\; \backslash cdots\; \backslash otimes\; V^\{\backslash otimes\; k\_n\}\; \backslash \; \backslash overset\{g\_1\; \backslash otimes\; \backslash cdots\; \backslash otimes\; g\_n\}\backslash longrightarrow\; \backslash \; V^\{\backslash otimes\; n\}\; \backslash \; \backslash overset\{f\}\backslash to\; \backslash \; V$,
3. (identity) The identity element in $\backslash mathcal\{End\}\_V(1)$ is the identity map $\backslash operatorname\{id\}\_V$,
4. (symmetric group action) $S\_n$ operates on $\backslash mathcal\{End\}\_V(n)$ by permuting the components of the tensors in $V^\{\backslash otimes\; n\}$.

If $\backslash mathcal\{O\}$ is an operad, a k-linear operad algebra over $\backslash mathcal\{O\}$ is given by a finite-dimensional vector space V over k and an operad morphism $\backslash mathcal\{O\}\; \backslash to\; \backslash mathcal\{End\}\_V$; this amounts to specifying concrete multilinear operations on V that behave like the operations of $\backslash mathcal\{O\}$. (Notice the analogy between operads&operad algebras and rings&modules: a module over a ring R is given by an abelian group M together with a ring homomorphism $R\; \backslash to\; \backslash operatorname\{End\}(M)$.)

Depending on applications, variations of the above are possible: for example, in algebraic topology, instead of vector spaces and tensor products between them, one uses (reasonable) topological spaces and cartesian products between them.

Image:Composition in the little discs operad.svgThe little 2-disks operad is a topological operad where $P(n)$ consists of ordered lists of n disjoint disks inside the unit disk of $\backslash R^2$ centered at the origin. The symmetric group acts on such configurations by permuting the list of little disks. The operadic composition for little disks is illustrated in the accompanying figure to the right, where an element $\backslash theta\backslash in\; P(3)$ is composed with an element $(\backslash theta\_1,\backslash theta\_2,\backslash theta\_3)\backslash in\; P(2)\backslash times\; P(3)\backslash times\; P(4)$ to yield the element $\backslash theta\; \backslash circ\; (\backslash theta\_1,\backslash theta\_2,\backslash theta\_3)\backslash in\; P(9)$ obtained by shrinking the configuration of $\backslash theta\_i$ and inserting it into the i-th disk of $\backslash theta$, for $i=1,2,3$.

Analogously, one can define the little n-disks operad by considering configurations of disjoint n-balls inside the unit ball of $\backslash R^n$.Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvily (2005) Geometric and Algebraic Topological Methods in Quantum Mechanics, {{isbn|981-256-129-3}}, [https://books.google.com/books?id=fLbisfrkWpoC&pg=PA474 pp. 474,475]

Originally the little n-cubes operad or the little intervals operad (initially called little n-cubes PROPs) was defined by Michael Boardman and Rainer Vogt in a similar way, in terms of configurations of disjoint axis-aligned n-dimensional hypercubes (n-dimensional intervals) inside the unit hypercube.{{Cite book|title=Axiomatic, Enriched and Motivic Homotopy Theory|last=Greenlees|first=J. P. C.|publisher=Springer Science & Business Media|year=2002|isbn=978-1-4020-1834-3|series=Proceedings of the NATO Advanced Study Institute on Axiomatic, Enriched and Motivic Homotopy Theory|location=Cambridge, United Kingdom|pages=154–156}} Later it was generalized by May{{cite journal | last1 = May | first1 = J. P. | year = 1977 | title = Infinite loop space theory | url = http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.bams/1183538891&page=record| journal = Bull. Amer. Math. Soc. | volume = 83 | issue = 4| pages = 456–494 | doi=10.1090/s0002-9904-1977-14318-8| doi-access = free }} to the little convex bodies operad, and "little disks" is a case of "folklore" derived from the "little convex bodies".{{Cite arXiv |eprint=math/9803156 |title=Grafting Boardman's Cherry Trees to Quantum Field Theory |last1=Stasheff |first1=Jim |year=1998}}

In graph theory, rooted trees form a natural operad. Here, $P(n)$ is the set of all rooted trees with n leaves, where the leaves are numbered from 1 to n. The group $S\_n$ operates on this set by permuting the leaf labels. Operadic composition $T\backslash circ\; (S\_1,\backslash ldots,S\_n)$ is given by replacing the i-th leaf of $T$ by the root of the i-th tree $S\_i$, for $i=1,\backslash ldots,n$, thus attaching the n trees to $T$ and forming a larger tree, whose root is taken to be the same as the root of $T$ and whose leaves are numbered in order.

Image:Swiss-cheese-operad.pdf

The Swiss-cheese operad is a two-colored{{Definition needed|date=February 2025}} topological operad defined in terms of configurations of disjoint n-dimensional disks inside a unit n-semidisk and n-dimensional semidisks, centered at the base of the unit semidisk and sitting inside of it. The operadic composition comes from gluing configurations of "little" disks inside the unit disk into the "little" disks in another unit semidisk and configurations of "little" disks and semidisks inside the unit semidisk into the other unit semidisk.

The Swiss-cheese operad was defined by Alexander A. Voronov.{{Cite book|title=The Swiss-cheese operad|last=Voronov|first=Alexander A.|publisher=AMS|year=1999|isbn=978-0-8218-7829-3|series=Contemporary Mathematics|location=Baltimore, Maryland, United States|pages=365–373}} It was used by Maxim Kontsevich to formulate a Swiss-cheese version of Deligne's conjecture on Hochschild cohomology.{{cite journal | last1 = Kontsevich | first1 = Maxim | year = 1999 | title = Operads and Motives in Deformation Quantization | url = https://link.springer.com/article/10.1023/A:1007555725247| journal = Lett. Math. Phys. | volume = 48 | pages = 35–72 | doi=10.1023/A:1007555725247 | arxiv = math/9904055 | bibcode = 1999math......4055K | s2cid = 16838440 }} Kontsevich's conjecture was proven partly by Po Hu, Igor Kriz, and Alexander A. Voronov{{cite journal | last1 = Hu | first1 = Po | last2 = Kriz | first2 = Igor | last3 = Voronov | first3 = Alexander A. | year = 2006 | title = On Kontsevich's Hochschild cohomology conjecture | journal = Compositio Mathematica | volume = 142 | issue = 1| pages = 143–168 | doi=10.1112/S0010437X05001521 | doi-broken-date = 2 March 2025 | doi-access = free | arxiv = math/0309369 }} and then fully by Justin Thomas.{{cite journal | last1 = Thomas | first1 = Justin | year = 2016 | title = Kontsevich's Swiss cheese conjecture | url = https://projecteuclid.org/euclid.gt/1510858920| journal = Geom. Topol. | volume = 20 | issue = 1| pages = 1–48 | doi = 10.2140/gt.2016.20.1 | arxiv = 1011.1635 | s2cid = 119320246 }}

Another class of examples of operads are those capturing the structures of algebraic structures, such as associative algebras, commutative algebras and Lie algebras. Each of these can be exhibited as a finitely presented operad, in each of these three generated by binary operations.

For example, the associative operad is a symmetric operad generated by a binary operation $\backslash psi$, subject only to the condition that

:$\backslash psi\backslash circ(\backslash psi,1)=\backslash psi\backslash circ(1,\backslash psi).$

This condition corresponds to associativity of the binary operation $\backslash psi$; writing $\backslash psi(a,b)$ multiplicatively, the above condition is $(ab)c\; =\; a(bc)$. This associativity of the operation should not be confused with associativity of composition which holds in any operad; see the axiom of associativity, above.

In the associative operad, each $P(n)$ is given by the symmetric group $S\_n$, on which $S\_n$ acts by right multiplication. The composite $\backslash sigma\; \backslash circ\; (\backslash tau\_1,\; \backslash dots,\; \backslash tau\_n)$ permutes its inputs in blocks according to $\backslash sigma$, and within blocks according to the appropriate $\backslash tau\_i$.

The algebras over the associative operad are precisely the semigroups: sets together with a single binary associative operation. The k-linear algebras over the associative operad are precisely the associative k-algebras.

The terminal symmetric operad is the operad which has a single n-ary operation for each n, with each $S\_n$ acting trivially. The algebras over this operad are the commutative semigroups; the k-linear algebras are the commutative associative k-algebras.

Similarly, there is a non-$\backslash Sigma$ operad for which each $P(n)$ is given by the Artin braid group $B\_n$. Moreover, this non-$\backslash Sigma$ operad has the structure of a braided operad, which generalizes the notion of an operad from symmetric to braid groups.

In linear algebra, real vector spaces can be considered to be algebras over the operad $\backslash R^\backslash infty$ of all linear combinations {{Citation needed|date=October 2022}}. This operad is defined by $\backslash R^\backslash infty(n)=\backslash R^n$ for $n\backslash in\backslash N$, with the obvious action of $S\_n$ permuting components, and composition $\backslash vec\{x\}\backslash circ\; (\backslash vec\{y\_1\},\backslash ldots,\backslash vec\{y\_n\})$ given by the concatentation of the vectors $x^\{(1)\}\backslash vec\{y\_1\},\backslash ldots,x^\{(n)\}\backslash vec\{y\_n\}$, where $\backslash vec\{x\}=(x^\{(1)\},\backslash ldots,\; x^\{(n)\})\backslash in\backslash R^n$. The vector $\backslash vec\{x\}=(2,3,-5,0,\backslash dots)$ for instance represents the operation of forming a linear combination with coefficients 2,3,-5,0,...

This point of view formalizes the notion that linear combinations are the most general sort of operation on a vector space – saying that a vector space is an algebra over the operad of linear combinations is precisely the statement that all possible algebraic operations in a vector space are linear combinations. The basic operations of vector addition and scalar multiplication are a generating set for the operad of all linear combinations, while the linear combinations operad canonically encodes all possible operations on a vector space.

Similarly, affine combinations, conical combinations, and convex combinations can be considered to correspond to the sub-operads where the terms of the vector $\backslash vec\{x\}$ sum to 1, the terms are all non-negative, or both, respectively. Graphically, these are the infinite affine hyperplane, the infinite hyper-octant, and the infinite simplex. This formalizes what is meant by $\backslash R^n$ being or the standard simplex being model spaces, and such observations as that every bounded convex polytope is the image of a simplex. Here suboperads correspond to more restricted operations and thus more general theories.

The commutative-ring operad is an operad whose algebras are the commutative rings. It is defined by $P(n)=\backslash Z[x\_1,\backslash ldots,x\_n]$, with the obvious action of $S\_n$ and operadic composition given by substituting polynomials (with renumbered variables) for variables. A similar operad can be defined whose algebras are the associative, commutative algebras over some fixed base field. The Koszul-dual of this operad is the Lie operad (whose algebras are the Lie algebras), and vice versa.

Typical algebraic constructions (e.g., free algebra construction) can be extended to operads. Let $\backslash mathbf\{Set\}^\{S\_n\}$ denote the category whose objects are sets on which the group $S\_n$ acts. Then there is a forgetful functor $\backslash mathsf\{Oper\}\; \backslash to\; \backslash prod\_\{n\backslash in\backslash N\}\; \backslash mathbf\{Set\}^\{S\_n\}$, which simply forgets the operadic composition. It is possible to construct a left adjoint $\backslash Gamma:\; \backslash prod\_\{n\backslash in\backslash N\}\; \backslash mathbf\{Set\}^\{S\_n\}\backslash to\; \backslash mathsf\{Oper\}$ to this forgetful functor (this is the usual definition of free functor). Given a collection of operations E, $\backslash Gamma(E)$ is the free operad on E.

Like a group or a ring, the free construction allows to express an operad in terms of generators and relations. By a free representation of an operad $\backslash mathcal\{O\}$, we mean writing $\backslash mathcal\{O\}$ as a quotient of a free operad $\backslash mathcal\{F\}\; =\; \backslash Gamma(E)$ where E describes generators of $\backslash mathcal\{O\}$ and the kernel of the epimorphism $\backslash mathcal\{F\}\; \backslash to\; \backslash mathcal\{O\}$ describes the relations.

A (symmetric) operad $\backslash mathcal\{O\}\; =\; \backslash \{\; \backslash mathcal\{O\}(n)\; \backslash \}$ is called quadratic if it has a free presentation such that $E\; =\; \backslash mathcal\{O\}(2)$ is the generator and the relation is contained in $\backslash Gamma(E)(3)$.{{cite journal | last1 = Markl | first1 = Martin| year = 2006 | title = Operads and PROPs | journal = Handbook of Algebra | volume = 5 | pages = 87–140| doi = 10.1016/S1570-7954(07)05002-4 | isbn = 9780444531018| s2cid = 3239126}} Definition 37

Clones are the special case of operads that are also closed under identifying arguments together ("reusing" some data). Clones can be equivalently defined as operads that are also a minion (or clonoid).

{{expand section|date=December 2018}}

In {{harvtxt|Stasheff|2004}}, Stasheff writes:

:Operads are particularly important and useful in categories with a good notion of "homotopy", where they play a key role in organizing hierarchies of higher homotopies.

• PRO (category theory)
• Algebra over an operad
• Higher-order operad
• E∞-operad
• Pseudoalgebra
• Multicategory

{{reflist|group=note}}

{{Reflist}}

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| title = Higher Operads, Higher Categories

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| arxiv = math/0305049

| bibcode = 2004hohc.book.....L

}}

• {{Cite book

| author = Martin Markl, Steve Shnider, Jim Stasheff

| year = 2002

| title = Operads in Algebra, Topology and Physics

| publisher = American Mathematical Society

| isbn = 978-0-8218-4362-8

| url = https://www.ams.org/bookstore?fn=20&arg1=survseries&item=SURV-96

}}

• {{Cite arXiv

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| first = Martin

| date = June 2006

| title = Operads and PROPs

| eprint = math/0601129

}}

• {{Cite journal

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| title = What Is...an Operad?

| journal = Notices of the American Mathematical Society

|date=June–July 2004

| volume = 51

| issue = 6

| pages =630–631

| url = https://www.ams.org/notices/200406/what-is.pdf

| access-date = 17 January 2008

}}

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• {{Citation | last=Zinbiel | first=Guillaume W. |author-link=Jean-Louis Loday| editor2-last=Guo | editor2-first=Li | editor1-last=Bai | editor1-first=Chengming | editor3-last=Loday | editor3-first=Jean-Louis | title=Operads and universal algebra | series = Nankai Series in Pure, Applied Mathematics and Theoretical Physics | isbn=9789814365116 | year=2012 | volume=9 | chapter=Encyclopedia of types of algebras 2010 | arxiv=1101.0267 | pages=217–298 | bibcode=2011arXiv1101.0267Z }}
• {{Citation|last=Fresse|first=Benoit|title=Homotopy of Operads and Grothendieck-Teichmüller Groups|series=Mathematical Surveys and Monographs|publisher=American Mathematical Society|isbn=978-1-4704-3480-9|mr=3643404|zbl=1373.55014|date=17 May 2017}}
• Miguel A. Mendéz (2015). Set Operads in Combinatorics and Computer Science. SpringerBriefs in Mathematics. {{ISBN|978-3-319-11712-6}}.
• Samuele Giraudo (2018). Nonsymmetric Operads in Combinatorics. Springer International Publishing. {{ISBN|978-3-030-02073-6}}.

• {{nlab|id=operad}}
• https://golem.ph.utexas.edu/category/2011/05/an_operadic_introduction_to_en.html

Category:Abstract algebra

Category:Category theory